Complexity in physical, living and mathematical systems

Abstract

This thesis is in a first place a compendium of different works addressing the emer¬gence of analogous complex behavior in areas of different garment, such as Social systems, Ecology, Networks, Stochastic Algorithms or Mathematical sequences. It also proposes in a second place some new methods and methodologies for Complex systems analysis. Some material is based on seven published papers and two preprints under review whose references can be found in the Concluding section. The work is divided in three general chapters, each of these having a specific introduction as well as a specific reference section. Chapter 1 studies the onset of complex behavior in living systems. Concretely, the first part of this chapter addresses the problem of hierarchy formation in societies as a collective induced phenomenon. Two different models of hierarchy formation are studied, namely the Bonabeau model and a modified version of the former one introduced by Stauffer. These agent based models present a phase transition distin¬guishing a phase where the system lacks hierarchy (the so called egalitarian regime) and a phase where hierarchy grows up sharply. Applying stability analysis techniques, we introduce new analytical developments and derive results in mean field approxi¬mation that agree with the numerics. In the second part of this chapter the task of designing ecological reserves with max¬imal biodiversity is studied from a probabilistic viewpoint. We analytically find the size distribution that maximizes biodiversity among a set of r reserves for a neutral case of uniform species colonization probability, and provide numerical simulations of the non neutral case. Chapter 2 embodies the study of different mathematical systems that evidence com¬plex behavior. In the first part we present a stochastic algorithm that generates primes. An algorithmic phase transition takes place distinguishing the ability of the algorithm to generate primes. Both Monte Carlo simulations and analytical developments are provided in order to characterize the dynamics of the system and explain what mechanisms are the responsible for the onset of the phase transition. Some connections between Computational complexity theory, Statistical physics, Number theory and Network theory are also outlined. In the same spirit, the second part of the chapter focuses on a simple algorithm whose dynamics evidence Self-Organized Criticality (SOC). We prove that this dynamical behavior is directly related to the underlying network of interactions of the system. Specifically, we find that that the underlying network's scale free topology induces criticality in the system's dynamics. We claim that this is a general mechanism for the onset of SOC. Finally, in the third part of this chapter we present an as yet unnoticed statistical pattern in both the prime number distribution and the Riemann zeta zero distribu¬tion. We prove that the nature of this pattern is a consequence of the prime number theorem. In Chapter 3 we gather some new methods and tools for Complex System analysis. In the first part we present the Self-Overlap method (SO), a method for the analysis of generic cooperative systems, and compare it with the well known Damage Spread¬ing method (DM). We claim that SO is computationally faster than DS, analytically simpler and lacks some of the ambiguities that DS evidences. We justify our claim by analyzing the thermodynamics and stability of the well known 2D Ising model through SO. In the second part of the chapter we introduce the Visibility graph, an algorithm that maps time series into networks and stands as a brand new tool for time series analysis. We present the method and some of its properties, explaining how Network theory can be used to describe the properties of time series. Finally, in the third part of the chapter we show that the Visibility algorithm stands as a new method to estimate the Hurst exponent of fractal series (namely fractional Brownian motion and /_/3 noises). Both numerical simulations and analytical developments are outlined, as well as some applications to real time series. We finally provide a concluding chapter that gathers the particular conclusions of each chapter as well as a list of the publications derived from this thesis.